The circle method and shifted convolution sums involving the divisor function

Abstract

Let $Q\left(x\right)$ be a positive definite integral quadratic form with the determinant D being squarefree, and $r(n,Q)$ denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function $d\left(n\right)$ and Fourier coefficients $r(n,Q)$. With more efforts, our method should have a number of applications for other multiplicative functions.